Power law convergence and concavity for the Logarithmic Schrödinger equation

2024/11/03 by Gallo, Marco, Mosconi, Sunra, Squassina, Marco · 2 citations
#26B25 #35B09 #35B53 #35B99 #35E10 #35J60 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2411.01614

Abstract

We study concavity properties of positive solutions to the Logarithmic Schrödinger equation -Δu=u log u² in a general convex domain with Dirichlet conditions. To this aim, we analyse the auxiliary Lane-Emden problems -Δu = σ (u^q-u) and build, for any σ>0 and q>1, solutions u_q such that u_q^(1-q)/2 is convex. By choosing σ_q=2/(1-q) and letting q → 1⁺ we eventually construct a solution u of the Logarithmic Schrödinger equation such that log u is concave. This seems one of the few attempts in studying concavity properties for superlinear, sign changing sources. To get the result, we both make inspections on the constant rank theorem and develop Liouville theorems on convex epigraphs, which might be useful in other frameworks.

Cited by

• On the ground state of the nonlinear Schrödinger equation: asymptotic behavior at the endpoint powers
• On the dependence of the nonlinear Schrodinger flow upon the power of the nonlinearity

Related

• Concavity properties for quasilinear equations and optimality remarks
• Log-concavity and strong log-concavity: a review
• Quasilinear Schrödinger equations with concave and convex nonlinearities
• Existence of a positive solution for a logarithmic Schrödinger equation with saddle-like potential
• Normalized solutions for logarithmic Schrödinger equation with a perturbation of power law nonlinearity